The document discusses the tractability of crisp representations of fuzzy description logics and their applicability in reasoning due to vagueness in knowledge representation. It evaluates different fuzzy logics, specifically Zadeh, Godel, and Łukasiewicz, and their closure under reduction conditions. The findings indicate that Zadeh fuzzy logic offers smaller crisp representations, while Godel and Łukasiewicz have varying requirements impacting their tractability.

1. Tractability of the Crisp Representations of
Tractable Fuzzy Description Logics
Fernando Bobillo
fbobillo@unizar.es
Department of Computer Science and Systems Engineering
University of Zaragoza, Spain
Joint research with M. Delgado (DECSAI, University of Granada, Spain)
URSW 2010
Shanghai (China)
November 2010
F. Bobillo (DIIS, Unizar) Representations of Tractable Fuzzy DLs URSW 2010 1/9

2. Introduction
Classical ontology languages are not appropriate to deal with
vagueness or imprecision in the knowledge.
Solution: Fuzzy Description Logics (DLs).
An important line of research is the computation of an equivalent
crisp representation of a fuzzy ontology.
This way, it is possible to reason with the obtained crisp ontology,
making it possible to reuse classical ontology languages, DL
reasoners, and other resources.
It is possible to reason with very expressive fuzzy DLs, and with
different fuzzy logics:
Zadeh
¨
Godel
Łukasiewicz
Our goal is to study some property (tractability) of the crisp
representations of fuzzy ontologies. Fuzzy DLs
F. Bobillo (DIIS, Unizar) Representations of Tractable URSW 2010 2/9

3. Tractable DLs
Expressive power compromised for the efficiency of reasoning.
The standard language OWL 2 has 3 fragments (profiles):
OWL 2 EL
OWL 2 QL
OWL 2 RL
Complexity:
OWL 2 EL, OWL 2 RL: polynomial time w.r.t. the ontology size.
OWL 2 QL: L OG S PACE w.r.t. the size of the ABox.
F. Bobillo (DIIS, Unizar) Representations of Tractable Fuzzy DLs URSW 2010 3/9

4. Tractable DLs
Relation of some OWL 2 constructors and its profiles:
OWL 2 OWL 2 EL OWL 2 QL OWL 2 RL
Class
ObjectIntersectionOf restricted
ObjectUnionOf restricted
ObjectComplementOf restricted restricted
ObjectAllValuesFrom restricted
ObjectSomeValuesFrom restricted restricted
DataAllValuesFrom restricted
DataSomeValuesFrom restricted
...
ObjectProperty
DatatypeProperty
...
ClassAssertion
ObjectPropertyAssertion
SubClassOf
SubObjectPropertyOf
SubDataPropertyOf
...
F. Bobillo (DIIS, Unizar) Representations of Tractable Fuzzy DLs URSW 2010 3/9

5. Motivation
Definition
A fuzzy DL language X is closed under reduction iff the crisp
representation of a fuzzy ontology in X is in the (crisp) DL language X .
Sometimes, fuzzy DL languages are closed under reduction.
The objective of this paper is to determine in a precise way when
this property holds, focusing on tractable fuzzy DLs.
F. Bobillo (DIIS, Unizar) Representations of Tractable Fuzzy DLs URSW 2010 4/9

6. The case of Zadeh fuzzy logic
Zadeh logic makes it possible to obtain smaller crisp
¨
representations than with Godel and Łukasiewicz logics.
Example:
From a : C D ≥ 0.6 we can deduce both a : C ≥ 0.6 and
a : D ≥ 0.6 .
In Łukasiewicz logic, this is not possible, and we have to build a
disjunction over all the possibilities.
From a : C D ≥ 0.6 , deduce a : C ≥ 1 and a : D ≥ 0.6 ,
or a : C ≥ 0.9 and a : D ≥ 0.7 ,
or a : C ≥ 0.8 and a : D ≥ 0.8 ,
or a : C ≥ 0.7 and a : D ≥ 0.9 ,
or a : C ≥ 0.6 and a : D ≥ 1 .
¨
In Godel implication, we have a similar problem.
F. Bobillo (DIIS, Unizar) Representations of Tractable Fuzzy DLs URSW 2010 5/9

7. The case of Zadeh fuzzy logic
Property
In Zadeh fuzzy logic, a fuzzy DL language X is closed under reduction
iff it includes GCIs and role hierarchies.
This result applies to OWL 2 EL, OWL 2 QL, and OWL 2 RL.
Example
Let us assume the language ALC.
Since ALC does not contain role hierarchies, the property fails.
Hence, fuzzy ALC is not closed under reduction.
This is intuitive, because the crisp representations contains role
hierarchies (R≥α R≥β ), which are not part of ALC.
F. Bobillo (DIIS, Unizar) Representations of Tractable Fuzzy DLs URSW 2010 5/9

8. ¨
The case of Godel fuzzy logic
Property
In Godel fuzzy logic, a fuzzy DL language X is closed under reduction
¨
iff it verifies each of the following conditions:
X includes GCIs.
X includes role hierarchies.
If X includes universal restrictions, then it also include conjunction.
This result applies to OWL 2 EL, OWL 2 QL, and OWL 2 RL.
F. Bobillo (DIIS, Unizar) Representations of Tractable Fuzzy DLs URSW 2010 6/9

9. The case of Łukasiewicz fuzzy logic
Property
In Łukasiewicz fuzzy logic, a fuzzy DL language X is not closed under
reduction if it verifies some of the following conditions:
X does not include GCIs.
X does not include role hierarchies.
X includes one and only one of disjunction and conjunction.
X includes existential restrictions, but not disjunction.
X includes universal restrictions, but not conjunction.
This result applies to OWL 2 EL, OWL 2 QL, and OWL 2 RL.
OWL 2 EL / OWL 2 QL support conjunction but not disjunction.
OWL 2 RL allows intersection as a superclass expression, but
it does not allow disjunction there.
We only have a partial result.
We only know a crisp representation for Łn ALCHOI.
F. Bobillo (DIIS, Unizar) Representations of Tractable Fuzzy DLs URSW 2010 7/9

10. Size of the crisp representations
¨
Zadeh and Godel OWL 2 QL:
Crisp representations are in crisp OWL 2 QL.
A crisp ontology with an ABox with the same size of the fuzzy one.
The complexity of reasoning depends on the number of assertions.
TBox and RBox are larger than the original fuzzy ones.
¨
Zadeh and Godel OWL 2 EL / OWL 2 RL:
Crisp representations are in crisp OWL 2 EL / RL.
TBox and RBox are larger than the original fuzzy ones.
Reasoning depends on the size of the ontology.
¨
Godel OWL 2 RL makes concept expressions larger than
Zadeh, because of universal restrictions.
¨
Godel OWL 2 EL / QL do not, since there are not universal
restrictions.
It is specially important to use optimized crisp representa-
tions (e.g., do not consider domain/range axioms as GCIs).
F. Bobillo (DIIS, Unizar) Representations of Tractable Fuzzy DLs URSW 2010 8/9

11. Comments?
Thank you very much for your attention
F. Bobillo (DIIS, Unizar) Representations of Tractable Fuzzy DLs URSW 2010 9/9